Words To Know Variable Expressions Vocabulary. Translating Words to Variable Expressions 1. The SUM...

Words To Know

Variable Expressions VocabularyVariable Expressions Vocabulary

Translating Words to Variable Translating Words to Variable ExpressionsExpressions

1. The SUM of a number and nine 2. The DIFFERENCE of a number and nine

3. The PRODUCT of a number and nine4. The QUOTIENT of a number and nine

5. One ninTH OF a number

n + 9 n – 9

9n

or

9. A number LESS THAN nine

–n 9

7. A number SQUARED

n2

6. Nine times THE QUANTITY OF a number increased by ten.

n3

8. A number CUBED

9(n + 10)

* When you see the phrase less than, reverse the terms.

Translating Variable ExpressionsTranslating Variable ExpressionsTranslate each mathematical expression into a verbal phrase

without using the words: “plus”, “add”, “minus”, “subtracted”, “”take–away”, multiplied”, “times”, “over”,

“power”, or “divided” ..

16. 13a

17.

18. y – 11

19. 3y + 8

20. 6 ÷ n2

21. 7(x + 1)

22. b3 – 4

the product of thirteen and a number

the quotient of fourteen and a number

the difference of a number and eleven, OR eleven less than a number OROR

a number decreased by eleven

eight more than the product of three and a number, OR the product of three and a number increased

by eight

the quotient of six and a number squared

seven, times the quantity of one more than a number, OR seven, times the quantity of a number increased

by onethe difference of a number cubed and four, OR a number cubed decreased

by four, OR four less than a number cubed

a

14

Simplifying Using Order of Operations

(1 + 3)2• 4

6 – 2 ÷ (–1)

Evaluate the

numerator and

denominator

separately( )

2• 4

6 –

4

• 4 646 – 2 ÷ (–1)

(–2)

+ 6 2

8

16

648

=8

–4 +[8 – (5 + 9)]•2Evaluate

inside the brackets

first...–4 +[8 – ( 14 )]•2

–4 +[ –6 ]•2

–4 + –12

–16

...then treat the brackets

like parenthesis

1. 2.

Evaluate Evaluate each expression each expression using:using:

x = –3 y = –2 z = 6x = –3 y = –2 z = 6

1) –1) –xx

––(–3)(–3) + + 33 33

2) –2) –yy

––(–2)(–2) + + 22 22

3) –3) –zz

––66 ––66

1. Substitute –3 for x only. 2. Leave the negative (–) in front of the x

alone.3. Now, simplify the signs (kill the sleeping

man)

1. Substitute –2 for y only. 2. Leave the negative (–) in front of the y

alone.3. Now, simplify the signs.

1. Substitute 6 for z only. 2. Leave the negative (–) in front of the z

alone.3. Now, simplify the signs.

Evaluating Variable Expressions with Negative Variables

4) 4) x – yx – y

––3 3 – – (–2)(–2) – –3 3 + 2+ 2 – –11

5) 5) x – zx – z

––3 3 – – 66 – –99

6) 6) z – xz – x

6 6 – – (–3)(–3) 6 6 + + 33 99

1. Substitute –3 for x , and –2 for y only. 2. Leave the subtraction sign (–) in front of the y

alone.3. Now, simplify the signs. (keep->change-

>change)4. Add the integers.

1. Substitute –3 for x , and 6 for z only. 2. Leave the subtraction sign (–) in front of the z

alone.3. Subtract the integers.

1. Substitute 6 for z , and –3 for x only. 2. Leave the subtraction sign (–) in front of the x

alone.3. Now, simplify the signs.4. Add the integers.

Evaluate Evaluate each expression each expression using:using:

x = –3 y = –2 z = 6x = –3 y = –2 z = 6


7) 7) xyxy

––3 3 •• (–2) (–2) 66

8) 8) yzyz

––2 2 •• 6 6 ––1212

9) 9) –xz–xz

––(–3) (–3) •• 6 6 ++3 • 63 • 6 1818

10) 10) –(–(xzxz))

––(((–3) (–3) •• 6 6)) ––((–18–18)) 1818

30) 30) yzyz

––3 3 – – 66 – –99

31) 31) z – xz – x

6 6 – – (–3)(–3) 6 6 + + 33 99

1. Substitute –3 for x , and –2 for y. 2. Multiply –– * Why? Two variables right next to each other.

1. Substitute –2 for y , and 6 for z. 2. Multiply –– * Why? Two variables right next to each other.

1. Substitute –3 for x , and 6 for z.2. Leave the negative sign in front of the x alone. 3. Simplify the signs.4. Multiply

1. Substitute –3 for x , and 6 for z.2. Leave the negative sign in front of the parenthesis, ( ), alone. 3. Multiply inside the parenthesis first.4. Simplify the signs.

Evaluate Evaluate each expression using: x = –3 y = –2 each expression using: x = –3 y = –2 z = 6 z = 6


11) 211) 2xx22

2 • 2 • (–3)(–3)22 2 • 2 • 9 9 1818

12) –212) –2xx22

– –2 • 2 • (–3)(–3)22 – –2 • 2 • 9 9 ––1818

13) 13) ((–2–2xx))22

((–2 • –2 • (–3)(–3)))22

( ( 6 6 ))22

3636

1. Substitute –3 for x.2. First, evaluate the exponent.3. Then, multiply. –– Why? When a number is right next

to a variable, multiply.

1. Substitute –3 for x.2. First, evaluate the exponent.3. Then, multiply.

1. Substitute –3 for x.2. First, evaluate inside parenthesis, ( ).3. Then, evaluate the exponent.



14) 214) 2xx33

2 • 2 • (–3)(–3)33 2 • 2 • (–27)(–27) ––5454

15) –215) –2xx33

– –2 • 2 • (–3)(–3)33 – –2 • 2 • (–27)(–27) 5454

16) 16) ((–2–2xx))33

((–2 • –2 • (–3)(–3)))33

( ( 6 6 ))33

216216

1. Substitute –3 for x.2. First, evaluate the exponent.

* Remember, (–3)(–3)33 is is (–3)(–3)••(–3)(–3)••(–3) = –27(–3) = –273. Then, multiply. –– Why? When a number is right next

to a variable, multiply.

1. Substitute –3 for x.2. First, evaluate the exponent.3. Then, multiply.

1. Substitute –3 for x.2. First, evaluate inside parenthesis, ( ).3. Then, evaluate the exponent.



Evaluating Variable ExpressionsEvaluating Variable Expressions

77 44

––2222

00 1111

2121 40405959

11..

5.5.4.4.

3.3.

2.2.

7.7.

6.6.

Simplifying Variable Expressions by Adding or Subtracting

––77a a + 11+ 11aa– – 99 ––33

Circle the variable terms, ...– – 12124a4a ... and box up the

constants Add the like terms.

1. 17a + a

Remember, aa = =

11aaso, put a

“1” in front of the a

2. –10 –7y + 6y – 3 3. 12b + 5 – 15 – 12b

–13 –1y

... or, get rid of the “1” 0 –

10

... or, get rid of the “0”

4. 14x + 7b – 9x + 19 – 11b – 21

– 4b + 5x – 2

5. 13 + 2(8 – g)Use

Distributive Property to get rid of

the parenthesis.

outer times first, then

outer times second

13 + 16 + 2g

29 + 2g

6. 13 +(– 19) – 6(n + 1) – 10n 13 +(– 19) – 6n – 6 – 10n

–12 – 16n

Simplifying Variable Expressions by Multiplication

8. 8. 7( –37( –3x x )) When you see constants (7 and –3) 7 and –3) and variables (x), (x), it’s easiest to simplify them separately.

77( ( –3–3xx )) First, multiply 7 7 and –33…

––2121 …then just bring down the xx(Why? It’s the only x x )xx

9. –199. –19aa • 10 • 10bcbc

––1919aa • • 1010bbcc When you see constants (–19 and 10) and multiple variables (a, b, and c), take it one at a time.

First, multiply –19 –19 and 1010…

…then bring down the a, b, and c

(Why? There’s only 1 of each.)––190190 abc

10. ( –1 )210. ( –1 )2yy 11. –511. –5aa( –5c )( –5c ) 12. (12. ( x x • • 8 )68 )6yy(–1(–1))22yy ––55aa(–5(–5cc)) ((xx • • 88))66yy

– –22yy 2525aacc 4848xxyy

Simplifying Variable Expressions Using the Distributive Property

–2(n +1)

When a number or variable term sits right next to terms inside

parenthesis, use the DistributiveDistributive PropertyProperty to simplify.

How?First, multiply the outer term, –2–2, by the 1st term in parenthesis, nn.

–2 •n –2 • +1

–2n –2 Then, multiply the outer term, –2, by the 2nd term in parenthesis, 1.

How to remember the Distributive How to remember the Distributive Property?Property?

“ “Outer times 1Outer times 1stst, then outer times , then outer times 22ndnd””

14. (9 – 6x)3 15. –4(8a + 7) 16. (–3p + 1)(–5) 17. 10(–c – 6)

27 – 18x –32a – 28 15p – 5 –10c – 60

13.

Are the bases, x, the same?

Multiplying Exponents Rule:Multiplying Exponents Rule:When When multiplyingmultiplying exponent exponent terms with like bases, keep terms with like bases, keep

the base, then the base, then addadd the the exponents.exponents.4 + 7 = 4 + 7 = So, we’re going to keep the base...

Are we multiplying or dividing the exponent terms?

18. Simplify 18. Simplify xx44 • • xx77

xx…then add the exponents

11 11

Rewrite.xx1111

19. a19. a66 • • aa99 20. b20. b • • bb55 21. y21. y • • yy44 • y • y44

a15

Careful:What’s

the invisible exponent over b ?

11

bb66 y9

Simplifying Variable Expressions by Multiplying ExponentsSimplifying Variable Expressions by Multiplying Exponents

GUIDED PRACTICESimplifying Variable Expressions by Multiplying ExponentsSimplifying Variable Expressions by Multiplying Exponents

22. 7x2 • 7x4 When you see both constants, 77, and variables, x, it’s easiest to simplify them separately.

SimplifySimplify

77 77 xx44xx22 ••

Let’s multiply the 7’s 7’s first...

4949 … then, multiply x2 • x4 .x6

23. 10y7 • 4y

10y7 • 4y

40y8

24. 3a5b • 3a6b8

Don’t panic:Just multiply

each part separately.

33aa55bb • • 33aa66bb88

99aa1111bb99

25. 2x5yz3 • yz

2x5yz3 • yz

2x5y2z4

Simplifying Variable Expressions by Dividing Exponents

7

12

x

x

Simplify

Are the bases, x, the same?

Are we multiplying or dividing the exponent terms?

Dividing Exponents Rule:Dividing Exponents Rule:When When dividingdividing exponent terms with like exponent terms with like bases, keep the base, then bases, keep the base, then subtractsubtract the the

exponents.exponents.So, we’re going to keep the base...

…then subtract the exponents

Rewrite.

12 – 7 = 12 – 7 =

xx5 5

nn–6–6

3

9

y

y27. 28. a4 ÷ a

29. 30.

8

9

b

b7n

n(huh?(huh?))

bbaa33yy66or 6

1

n

xx55

26.

Simplifying Variable Expressions by Dividing Exponents

31. Simplify

3

9

6

12

y

yWhen you see both constants, 12 and 612 and 6, and variables, y, it’s easiest to simplify them separately.

6

12Let’s simplify the fraction first...

… then, divide y9 and y3 .2

y6

34. 33. 32. a

a

9

45 2

6

8

10

32

n

n

6

12

8

12

15

5

p

p 35. 7

25

4

28

xy

zxy 36. 3

2

7

23

f

dd

5a

5

16 2n3

4p2

27

y

z3

3

7

23

f

d

Hint: The rest of the answers are fractions.

Words To Know Variable Expressions Vocabulary. Translating Words to Variable Expressions 1. The SUM...

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